This is a preliminary math question from the 2024 Xi'an Jiaotong University Junior Class Admissions, which is relatively simple, and the questions are as follows:
Topic: If the sunlight is at an angle of 60° to the horizontal plane, and the shadow length of the ball in the sunlight is 10 3cm, then the diameter of the ball is cm
From the title we know that the edge of the sun will be tangent to the ball, as shown below:
To solve geometry problems, the first thing is to make auxiliary lines, and the second is to grasp the properties of circular tangents.
Then we know that there are three tangent lines ab, af and bd in this problem, and we know that oe, of, and od are perpendicular to ab, af, and bd respectively from the tangent theorem, and here we need to find the diameter, so we can find the radius r.
The tangent length theorem shows that two tangents are drawn at a point outside the circle, and their tangents are of equal length, that is, ae=af, bd=be, and the tangent angles are bisected, so oae=30°
Since ame=30°, we can get eod=60°, bisecting the included angle, so eob=30°
Well, knowing these angles, it's easy to get.
tan∠oae=tan30°=r/ae=√3/3And so you goae+eb=√3*r+√3/3*r=10√3And so you gor=7.5, so the diameter isIt is important to see the nature of the geometric figure, which is directly related to the speed and accuracy of the problem.tan∠eob=tan30°=eb/r=√3/3
Through this question, we will expand some relevant knowledge points about circles, and draw pictures to make everyone look more clear and intuitive.
Circumferential angle theorem, as shown in the figure below:
The circumferential angle of an arc is half of the central angle of the circle, and the circumferential angle of the same arc is equal.
The theorem of the tangent angle of the string, as shown in the figure below:
Chord chamfer: The tangent point is on the circumference of the circle, one side is tangent to the circle, and the other side intersects the circle.
The magnitude of the chord chamfer is equal to the magnitude of the circumferential angle corresponding to the arc being clamped.
The cutting line theorem, as shown in the figure below: